On reliability of prediction in linear models

For a linear noisy system assume an estimate
\hat{\beta}
of the unknown parameter vector was obtained from a certain past data on inputs and outputs of the system. The prediction of the output for any input vector
x
is a linear combination of
x
and
\hat{\beta}
. The question of concern is: For what - values of the inputs would the error in their predicted outputs be the smallest? It is found that they are the inputs which lie on the fitted plane of the orthogonal regression among the past data of the clean inputs.